
where \(|\epsilon _{v_i}| \rightarrow 0\)  as \(\eta \rightarrow 0\) .
From the fact that \(({\mathbf {u}}^*, {\mathbf {w}}^*, {\mathbf {v}}^*)\)  is a fixed point of the classical WMMSE iteration [1]}, it follows that the first term in the right-hand side of (REF ) is equal to \(v_i^*\) .
This implies that the function \(g_k({\mathbf {H}}, {\mathbf {v}}^* + \epsilon ; {\mathbf {b}}^{(k)})\)  that we were looking for is given, elementwise, by the second and third terms in the right-hand side of (REF ).
Since \(\epsilon _{v_i} \rightarrow 0\)  as \(\eta \rightarrow 0\)  (equivalently, as \(k \rightarrow \infty \) ), it must be that the last term in (REF ) goes to 0 as \(k\)  goes to infinity, from where our result follows.
